NSolve函数无法求出含逆累积分布函数方程的数值解

Question

a = 2000; b = 500;

In[48]:= f[x_] := PDF[NormalDistribution[a, b], x] /; x > 0

In[49]:= F[y_] := CDF[NormalDistribution[a, b], y] /; y > 0

In[50]:= G[z_] := InverseCDF[NormalDistribution[a, b], z] /; 0 < z < 1

In[56]:= NSolve[\!\(

\*SubsuperscriptBox[\(\[Integral]\), \(G[

      z]\), \(+\[Infinity]\)]\(\((x - G[z])\) f[

      x] \[DifferentialD]x\)\) - z/f[G[z]] == 0, z]

\:6B63\:5728\:8BA1\:7B97In[56]:= NSolve::nsmet: 无法利用 NSolve 现有的方法求解该系统.

Out[56]= NSolve[

 ConditionalExpression[

  250 E^(-((-2000 + (-500 (-4 + Sqrt[2] InverseErfc[2 z]) && 

          0 < z < 1))^2/500000)) Sqrt[2/\[Pi]] - 

    500 E^((-2000 + (2000 - 500 Sqrt[2] InverseErfc[2 z] && 

         0 < z < 1))^2/500000) Sqrt[2 \[Pi]] z - 

    1/2 (-2000 + (-500 (-4 + Sqrt[2] InverseErfc[2 z]) && 

         0 < z < 1)) Erfc[(-2000 + (-500 (-4 + 

            Sqrt[2] InverseErfc[2 z]) && 0 < z < 1))/(500 Sqrt[2])] ==

    0, 0 <= z <= 1], z]

Answer 1

a = 2000;
b = 500;
f[x_] = PDF[NormalDistribution[a, b], x];
F[y_] = CDF[NormalDistribution[a, b], y];
G[z_] = Assuming[0 <= z <= 1, 
   Refine@InverseCDF[NormalDistribution[a, b], z]];
eqn = Integrate[(x - G[z]) f[x], {x, G[z], Infinity}] - z/f[G[z]];
Plot[eqn, {z, 0, 1}]
FindRoot[eqn == 0, {z, 0.5}]